Numerical material testing based on homogenization method for non-periodic media (Formulation using microscopic fluctuation displacement and macroscopic strain)

Abstract

We propose a method for numerical material testing that does not explicitly assume boundary conditions in homogenization analyses for microstructures with random or localized inhomogeneities. Specifically, instead of the periodic constraint condition on the fluctuation displacement in the RVE, a condition is imposed such that the domain integral of the gradient is zero, which preserves the definition of macroscopic strain in the homogenization theory while also suppressing rigid body rotation. Then, a scale-separated variational problem is defined with this condition introduced as a constraint. Next, the governing equations of the two-variable boundary value problem (BVP) are derived by applying Lagrange multiplier method to this variational problem. The macroscopic BVP is degenerated to a material point, and the microscopic BVP is transformed into a form specialized for numerical material testing and then discretized by the finite element method (FEM). The discretized equations are extended by adding the degrees of freedom corresponding to macroscopic stress and strain. Since the proposed method does not require any restrictions on the external shape of the microstructure and the extraction of external boundaries, it is relatively easy to implement into the FEM. Several numerical examples are presented to demonstrate the validity of the proposed by comparing the results with those of the conventional approach.